The University of New South Wales
School of Electrical Engineering and
Telecommunications
ELEC4632 Computer Control Systems
Final Examination, Term 3, 2021
1. Time allowed : 2 hours
2. Reading time : 10 minutes
3. Scanning time : 20 minutes
4. Total exam duration: 9:30am – 12:00pm (Sydney time).
5. The paper contains 6 questions (60 marks in total).
6. Each question has the value indicated.
7. This exam contributes 60% to the course final mark.
8. Candidates should attempt all questions.
9. This is an open book examination.
10. This paper contains 4 pages.
11. Please combine answers in one pdf or jpg file, name the file by
zID first name family name, and click the submit button before
the due time.
1
NO collaborations allowed and the work submitted must be
your own. During the exam you may NOT be in contact with
anyone else via any form of communication media (email, messag-
ing, phone, video conferencing, etc). Violating this will be consid-
ered an academic misconduct and disciplinary action will be taken
against anyone who is proven to have violated this rule.
ANSWERS MUST BE WRITTEN IN INK. EXCEPT WHERE THEY
ARE EXPRESSLY REQUIRED, PENCILS MAY ONLY BE USED FOR
DRAWING, SKETCHING OR GRAPHICAL WORK.
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Question 1 (10 marks)
Sample the continuous-time system
x˙(t) =
 3 0 07 0 −2
0 0 0
x(t) +
 0−2
1
u(t− 0.85),
y(t) =
(
−3 4 −2
)
x(t)
using the sampling interval h = 0.2.
Question 2 (10 marks)
A discrete-time control system is described by
x(k + 1) =
 2 5a −40 −3 a2 − 49
−3a −5 a− 4
x(k) +
 1− a
2 −2 0
0 a + 7 0
−3 3a a + 1
u(k)
where the parameter a varies from −∞ to +∞. Determine for which values
of the parameter a this system is
(a) reachable;
(b) controllable.
Question 3 (10 marks)
Given the system
x(k + 1) =
(
0.8 −0.6
1.4 −0.7
)
x(k) +
( −1
1
)
u(k),
y(k) =
(
2 −1
)
x(k).
Design deadbeat output feedback control law.
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Question 4 (10 marks)
The characteristic equation of a discrete-time control system is given by
z2 + K2z + K = 0
where the parameter K varies from −∞ to +∞. Determine the range of the
parameter K for stability.
Question 5 (10 marks)
Consider the following nonlinear discrete-time system
x(k + 1) = x(k) + x(k)2 + y(k)2,
y(k + 1) = −x(k)2 + y(k)− y(k)2.
(a) Is this system globally asymptotically stable?
(b) Is the singular point (0, 0) of this system asymptotically stable?
(c) Is the singular point (0, 0) of this system stable in the sense of
Lyapunov?
Question 6 (10 marks)
Consider the optimal control problem for the system
x(k + 1) = −2x(k) + u(k)
with initial condition x(0) = 1.5. Determine the optimal control strategy
and the optimal value of the cost function for the cost function
(a) J =
∞∑
k=0
u2(k)→ min;
(b) J =
50∑
k=0
u2(k)→ min .
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